Numeration in Laurent Series over Finite Fields

Christol`s theorem establishes a connection between the algebraicity of a Laurent series and the automaticity of its sequence of coefficients. It is conjectured that Christol`s theorem can be generalized to beta expansions over finite fields as well as other concepts of numeration. We want to get algebraicity results for this class by combining results from automata theory with number theoretic results for finite fields. So far, mainly finite state automata have been used in connection with number systems. For instance, they can encode the language of admissible expansions or perform additions with fixed numbers. In our project we define an new class of number systems with several bases. In this case the transducer automata generalize to cellular automata. Recently, expansions w.r.t. rational numbers like the 3/2 number system became an interesting topic of research. In the context of finite field numeration the analogue of these number systems are number systems w.r.t. non-monic polynomials as bases. Although, as in the rational case, these expansions are rather awkward, structural results on these expansions could be established. Interestingly, again cellular automata play a role in this context.

One focus of the project were digit systems in function fields as well as p-adic numbers. In general, problems for function fields are easier to solve than their counterparts in number fields. For function fields, it was possible for several types of digit systems to characterise algebraic elements by automaticity of their digit expansions. This is an analogy to the famous result of Christol. Corresponding results in are currently unknown.Another issue were topological and dynamical properties of Rauzyfractals. Results on dimension and symmetry of Rauzyfractals have been proved.Tent maps are continuous composites of two linear functions that act on the unit interval. The authors analyse a connection between dynamical systems induced by tent maps and the dynamics induced by a certain type of beta-expansion. Some new results on the set of periodic points could be established.